Octahedron after Sydler

3D Model

An irregular non-convex octahedron with triangular faces.

All but one of its dihedral angles are right-angles; the exception is the long edge, whose dihedral angle is arccos(-1/3) ≈ 109.5°.

J-P Sydler described a family of hexahedra in which all but two of the dihedral angles are right-angles (Conditions nécessaires et suffisantes pour l’équivalence des polyèdres de l’espace euclidien à trois dimensions, Commentarii Mathematici Helvetici, Volumen 40, 1965, Chapitre 1, Figure 2).

One member of Sydler’s family has its two non-right dihedral angles being 135° and arccos(-1/3). Gluing two of them together gives this octahedron.